Equivariant embedding of metrizable $G$-spaces in linear $G$-spaces

Given a Lie group $G$ we study the class $\mathcal{M}_G$ of proper metrizable $G$-spaces with metrizable orbit spaces, and show that any $G$-space $X \in \mathcal{M}_G$ admits a closed $G$-embedding into a convex $G$-subset $C$ of some locally convex linear $G$-space, such that $X$ has some $G$-neighborhood in $C$ which belongs to the class $\mathcal{M}_G$. As a corollary we see that any $G$-ANR for $\mathcal{M}_G$ is a $G$-ANE for $\mathcal{M}_G$.